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Equações diferenciais estocásticas e aplicações em R
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Abstract(s)
A presente dissertação aborda o tema sobre as equações diferenciais estocásticas e suas
aplicações em R, com recurso às técnicas computacionais. A abordagem teórica baseia-se nas
equações diferenciais estocásticas derivadas dos processos estocásticos. Neste, apresenta-se,
também, o teorema de Itô, que se revela fundamental para o alcance dos objectivos
pretendidos. Importa, portanto, realçar que o teorema de Itô facilitou a implementação prática
dos modelos usados para aplicação das equações diferenciais tomadas como exemplo em
finanças, através de algorítmos computacionalmente escritos no software R. O trabalho baseiase, ainda, no processo de Wiener, em que foi abordado o conteúdo sobre as equações
diferenciais estocásticas que serviram de suporte para a abordagem do cálculo de Itô.
Outrossim, houve a aplicação computacional do software R, nos packages “sde” e
“mixedsde”. E, entre vários, o modelo de Black-Scholes foi usado como um dos exemplos a
considerar na aplicação das equações diferenciais estocásticas em finanças, em que foram
feitos cálculos manuais e computacionais. Nesta ordem de ideia, sobre aplicação das equações
diferenciais estocásticas em R, o package “sde” foi explorado a partir de uma das suas funções
- o “sde.sim” - baseado na simulação de equações diferenciais estocásticas, com um interface
de aplicação de diferentes métodos e modelos de simulação, e o package “mixedsde”foi
explorado através de uma das suas várias funções - o “mixedsde.sim” - baseado na geração de
trajectórias de processos estocásticas, usando alguns dos seus modelos.
This dissertation work addresses the topic of stochastic differential equations, and their applications in R, using computational techniques. The theoretical approach based on stochastic differential equations, derived from stochastic processes. In this one, it is also presented the theorem of Itô, which is essential to achieve the objectives of the work. It is noteworthy that the theorem of Itô, facilitated the practical implementation of the models used for the application of differential equations, taking as an example in finance through algorithms computationally written in R software. The work is also based on the Wiener process, in which the content about the stochastic differential equations was approached, that served for the support of the Itô calculus approach, and to finish, the computational application of the R software, in the packages “sde” and “mixedsde”, in which the Black-Scholes model was used as one of the examples, in several models for considering, in the application of the stochastic differential equations in finances, in which manual and computational calculations were made. Thus, regarding the application of stochastic differential equations in R, the package “sde” was explored, through one of its functions, sde.sim, based on the simulation of stochastic differential equations, with an interface of application of different methods, and simulation models, and the package “mixedsde” was explored through one of its several functions, “mixedsde.sim”, based on the generation of stochastic process trajectories, using some of its models.
This dissertation work addresses the topic of stochastic differential equations, and their applications in R, using computational techniques. The theoretical approach based on stochastic differential equations, derived from stochastic processes. In this one, it is also presented the theorem of Itô, which is essential to achieve the objectives of the work. It is noteworthy that the theorem of Itô, facilitated the practical implementation of the models used for the application of differential equations, taking as an example in finance through algorithms computationally written in R software. The work is also based on the Wiener process, in which the content about the stochastic differential equations was approached, that served for the support of the Itô calculus approach, and to finish, the computational application of the R software, in the packages “sde” and “mixedsde”, in which the Black-Scholes model was used as one of the examples, in several models for considering, in the application of the stochastic differential equations in finances, in which manual and computational calculations were made. Thus, regarding the application of stochastic differential equations in R, the package “sde” was explored, through one of its functions, sde.sim, based on the simulation of stochastic differential equations, with an interface of application of different methods, and simulation models, and the package “mixedsde” was explored through one of its several functions, “mixedsde.sim”, based on the generation of stochastic process trajectories, using some of its models.
Description
Keywords
Equação diferencial estocástica Teorema de Itô Software R Packages “sde” Packages "sde.sim" Packages "mixedsde" Stochastic differential equation Itô's theorem
Citation
General, Ângelo Rafael - Equações diferenciais estocásticas e aplicações em R [Em linha]. [S.l.]: [s.n.]. 2024. 60 p.